Description

Problem 1
(10 points)
Use substitution to evaluate the following integrals.
a) (2 points)
b) (2 points)
Use integration by parts to evaluate the following integrals.
c) R cos(x)ln(sinx)dx (3 points)
d) (3 points)
Problem 2
(10 points)
a) Prove the reduction formula

(5 points) Hint: Use integration by parts and the fact that cos2(x) + sin2(x) = 1.
b) Suppose that f : R→R continuous and odd, i.e., satisfies −f(x) = f(−x). Show that
.
(5 points)
Problem 3
(10 points + 5 bonus points)
Using 2a)

Z
a) Evaluate cos2(x)dx. (4 points)
Z
b) Evaluate cos3(x)dx. (3 points)
c) Evaluate . (3 points)
Bonus:
Find the area between the curves x = 1 − y2 and y = −2x − 1. (5 bonus points)
Hint: At the end, integrate with respect to y not x.